David Mertens

# NAME

PDL::Fit::ExpRate - fitting noisy exponential decay

# VERSION

This document describes PDL::Fit::ExpRate version 0.02

# SYNOPSIS

`````` use PDL;
use PDL::Fit::ExpRate;

# Load x/y data, or generate some noisy stuff:
my \$xs = sequence(100)/10;
my (\$A, \$B, \$tau) = (5, 4, -10);
my \$ys = \$A + \$B * exp(\$xs / \$tau) + \$xs->grandom;

# Extract the parameters
my (\$fit_A, \$fit_b, \$fit_tau) = fit_exp_rate(\$xs, \$ys);

print "A was \$A; fit as \$fit_A\n";
print "B was \$B; fit as \$fit_B\n";
print "tau was \$tau; fit as \$fit_tau\n";

# Other useful functions

# Solve for \$coefs in \$A * \$coefs = \$y
# where \$A is 3x3
my \$A = pdl q[1 0 0 ; 0 2 0 ; 0 0 3];
my \$y = pdl q[5     ,   -4  ,     6];
my \$coefs = three_by_three_Householder(\$A, \$y);

# Perform a quadratic fit
# y = coefs[0] + coefs[1] * x + coefs[2] * x**2
my \$coefs = fit_quadratic(\$xs, \$ys);

# and more yet to be documented...``````

# DESCRIPTION

This module provides a handy function for fitting time series data to a single exponential growth or decay. A number of methods used by this method are also made available, including a method for fitting data to a second-order polynomial, and performing Householder diagonalization on 3x3 matrix systems, and various parameter estimation and single-step functions.

The naive approach to fitting data to an exponential decay involves taking the logarithm of the y-values and fitting the resulting transformed data to a line. This has a number of drawbacks. First, it leads to a bias in the errors: errors in the data associated with smaller y-values receive a stronger weight than those associated with larger y-values. Second, there is no simple way to handle an additive offset such as A in `y = A + B * e**(x/tau)`. Third and most important, the transformation technique breaks down if the noise leads to any data values that are negative. The only simple way to handle such data is to discard it. An approach involving a logarithmic transformation is biased and can only be used under limited circumstances.

The fitting method supplied by this module, "fit_exp_rate", takes a different and much more robust approach that does not suffer any of those errors. It performs a least-squares fit to an exponential directly to the data using an iterative guarded Newton method. The technique gives accurate results, is highly configurable, and converges rapidly. As an added bonus, the underlying C functions can be accessed by other PDL modules using PDL::PP::Include::Fit::ExpRate.

## three_by_three_Householder

Given A and y, solves A * coef = y

This function performs Householder diagonalization on a 3x3 matrix A with the given "answer" y. It effectively inverts the matrix, leading to a solution for coef that can be effectively written as:

`` coef = A**-1 * y``

where A**-1 is the matrix inverse of A and multiplication is matrix multiplication.

Here's an exampe of how to use it. I've deliberately made the A matrix simple so that you can perform the inversion in your head, but A need not be sparse.

`````` my \$A = pdl q[1 0 0 ; 0 2 0 ; 0 0 3];
my \$y = pdl q[5     ,   -4  ,     6];
my \$coefs = three_by_three_Householder(\$A, \$y);
my \$expected = pdl(5, -2, 2);
print "Got \$coefs; expected \$expected\n";``````

## fit_quadratic

Given x and y data, determines A, B, and C for y = A + B x + C x**2

This function determines the least-squares coefficients for a quadratic fit to a given set of x-y data. This is a simple PDL wrapper for the internal C function `exprate_fit_quadratic`. This usage is pretty simple:

`````` # Make some noisy data
my \$xs = sequence(50);
my \$ys = 5 + 3 * \$x + 4 * \$x**2 + grandom(\$xs);

# should be (5, 3, 4)
my \$coefs = fit_quadratic(\$xs, \$ys);``````

# SEE ALSO

There are a number of PDL modules for fitting data, including PDL::Fit::Linfit for linear fitting and PDL::Fit::Polynomial for fitting data to polynomials. Both of those modules depend on Slatec (though they may be installed on your system even if you don't have Slatec available). PDL::Fit::LM provides Levenberg-Marquardt fitting, and PDL::Fit::Gaussian provides methods for fitting a set of data to a normal (Gaussian) distribution.

# AUTHOR

David Mertens <dcmertens.perl@gmail.com>

# LICENCE AND COPYRIGHT

Documentation is copyright (c) 2012, David Mertens, all rights reserved.

Code is copyright (c) 2012, Northwestern University, all rights reserved.

This module is free software; you can redistribute it and/or modify it under the same terms as Perl itself. See perlartistic.